We consider a power series $f = \lambda I + F$ with $F \in O_2, \rho(F) > 0$. We still assume that $\lambda$ has modulus 1 and is not a root of unity. We consider the real $r_0 > 0$ given by question (24) (applied for $m = 1$) and the sequence $r_k$ defined by recursion from $r_0$ by the relation $$r_{k+1} = (1 - \alpha_{2^k})(1 + \alpha_{2^k}^2)^{-1}(1 + \alpha_{2^k})^{-1} \gamma_{2^k} r_k.$$ Show that there exist sequences $F_k$ and $P_k$ of elements of $O_2$, defined for $k \geqslant 0$, such that $F_0 = F$ and, for all $k \geqslant 0$, $$\begin{aligned}
& \lambda I + F_{k+1} = (I + P_k)^\dagger \circ (\lambda I + F_k) \circ (I + P_k), \\
& F_k \in O_{1+2^k}, \quad P_k \in O_{1+2^k}, \\
& \widehat{F_k}(r_k) \leqslant r_k, \quad \widehat{P_k}(r_{k+1}) \leqslant r_k - r_{k+1}.
\end{aligned}$$
We consider a power series $f = \lambda I + F$ with $F \in O_2, \rho(F) > 0$. We still assume that $\lambda$ has modulus 1 and is not a root of unity. We consider the real $r_0 > 0$ given by question (24) (applied for $m = 1$) and the sequence $r_k$ defined by recursion from $r_0$ by the relation
$$r_{k+1} = (1 - \alpha_{2^k})(1 + \alpha_{2^k}^2)^{-1}(1 + \alpha_{2^k})^{-1} \gamma_{2^k} r_k.$$
Show that there exist sequences $F_k$ and $P_k$ of elements of $O_2$, defined for $k \geqslant 0$, such that $F_0 = F$ and, for all $k \geqslant 0$,
$$\begin{aligned}
& \lambda I + F_{k+1} = (I + P_k)^\dagger \circ (\lambda I + F_k) \circ (I + P_k), \\
& F_k \in O_{1+2^k}, \quad P_k \in O_{1+2^k}, \\
& \widehat{F_k}(r_k) \leqslant r_k, \quad \widehat{P_k}(r_{k+1}) \leqslant r_k - r_{k+1}.
\end{aligned}$$